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Planck units: the natural system of measurement defined by fundamental constants

Planck units are a system of natural units derived from universal physical constants. They simplify equations in theoretical physics and provide characteristic scales for length, time, mass and temperature.

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Overview

Planck units form a system of measurement in which a small set of universal physical constants are assigned the numerical value 1. Introduced by Max Planck at the end of the 19th century, this system is designed so that the basic constants that characterize space, time, quantum action and thermal energy serve as the foundation for all derived units. Using Planck units can remove arbitrary human scales and make relationships among physical laws more transparent, particularly in domains that combine relativity, quantum mechanics and statistical physics. For a basic introduction to measurement systems see units of measurement and for biographical context on the originator see Max Planck.

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Max Planck 1933
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Defining constants and derived units

The Planck system is constructed from four primary constants that appear across fundamental theories. These are the speed of light in vacuum (c), the gravitational constant (G), the reduced Planck constant (ħ) and the Boltzmann constant (kB). When these constants are set equal to unity, the remaining physical quantities acquire characteristic Planck scales. Some of the most commonly cited derived Planck quantities include:

  • Planck length (ℓP)
  • Planck time (tP)
  • Planck mass (mP)
  • Planck temperature (TP)

Because the construction relies only on fundamental properties of empty space and constants of nature, Planck units are a form of natural units. They should be contrasted with human-defined standards such as the metre and second in the SI system.

Why these constants?

Each of the constants chosen for Planck units has a central role in a major physical theory. The constant c appears in special relativity, G is central to general relativity and classical gravity (Newton's law), ħ characterizes quantum mechanics (quantum theory), and kB links microstates to thermodynamic temperature (statistical mechanics, thermodynamics). Other constants, such as the vacuum permittivity ε0, play roles in electromagnetic units but are not required to set the Planck scale.

History and perspective

Planck proposed his unit system near the turn of the 20th century as part of his work on black-body radiation and the emerging quantum hypothesis. The idea of using only universal constants to define units is older than many modern unit systems and reflects a desire to base measurement on the physical structure of the universe rather than on local artifacts. Alternative natural-unit schemes exist (other systems), and different fields sometimes adopt convenient conventions for calculation, but the Planck system is widely discussed because it brings together gravity, quantum mechanics and thermodynamics.

Uses, importance and examples

In theoretical physics, expressing equations in Planck units frequently removes repetitive constants and clarifies which quantities are dimensionless or physically fundamental. This simplification is useful when exploring regimes where quantum and gravitational effects both matter, such as in early-universe cosmology or studies of quantum gravity. The Planck length and Planck time are often described as characteristic scales below which classical concepts of space and time may need revision; likewise Planck mass and Planck temperature mark natural crossover scales linking quantum, gravitational and thermal phenomena. Because they eliminate anthropocentric choices of scale, Planck units have occasionally been proposed as a neutral language for interstellar communication (communication with extraterrestrial intelligence), though practical messages typically combine many conventions. More prosaically, they are a bookkeeping tool that highlights dimensionless ratios and hierarchy problems noted by theorists such as Frank Wilczek, who emphasized how natural units change the framing of certain questions about particle masses and force strengths (algebraic simplifications, physical law).

Practical notes and distinctions

Planck units are not intended as replacements for measurement systems used in everyday science, engineering or commerce; SI remains the international standard. Instead, Planck units are most useful in conceptual and theoretical work. They are not the only natural unit choice, and some physical contexts favour other normalizations. Finally, because Planck-derived scales are extremely remote from everyday experience they serve primarily as a guide to where known theories may need new physics rather than as immediately measurable standards. For concise overviews and complementary discussions, see introductory materials on simplifying equations, the nature of the vacuum, and further reading collections at the defining constants and other resources (units, history).

Further reading and resources

Definitions

Basic sizes

The Planck units result from a simple dimension consideration. They result as mathematical expressions of the dimension of a length, time and mass, respectively, which contain only products and quotients of suitable powers of G, cand \hbar . If one additionally uses the electric permittivity of the vacuum ε \varepsilon _{0}and the Boltzmann constant k_{\mathrm {B} }, then a Planck charge and a Planck temperature can also be determined as further basic quantities. The Planck charge satisfies the condition that the gravitational force between two Planck masses and the electromagnetic force between two Planck charges are equally strong: {\displaystyle m_{\mathrm {P} }^{2}G/\;\!l_{\mathrm {P} }^{2}=q_{\mathrm {P} }^{2}/4\pi \varepsilon _{0}\,l_{\mathrm {P} }^{2}}.

Name

Size

Dimension

Term

Value in SI units

In other units

Planck length

Length

L

l_\mathrm{P} = \sqrt{\frac{\hbar\,G}{c^3}}

1,616 255(18) · 10−35 m

3,054 · 10−25 a0

Planck mass

Mass

M

m_\mathrm{P} = \sqrt{\frac{\hbar\,c}{G}}

2.176 434(24) - 10-8 kg

1.311 - 1019 u,
1.221 - 1019 GeV/c2

Planck time

Time

T

{\displaystyle \!\,t_{\mathrm {P} }={\sqrt {\frac {\hbar \,G}{c^{5}}}}={\frac {l_{\mathrm {P} }}{c}}}

5,391 247(60) · 10−44 s

Planck temperature

Temperature

Θ

{\displaystyle \!\,T_{\mathrm {P} }={\frac {m_{\mathrm {P} }\,c^{2}}{k_{\mathrm {B} }}}}

1,416 784(16) · 1032 K

Planck charge

Cargo

Q

{\displaystyle q_{\mathrm {P} }={\sqrt {4\pi \varepsilon _{0}\ \hbar \,c}}}

1,875 545 956(41) · 10−18 C

11,71 e

The formula symbols mean:

Instead of \,Gsometimes \,8\pi Gset to one, then the mass unit is the reduced Planck mass:

{\displaystyle {\overline {m_{\mathrm {P} }}}={\sqrt {\frac {\hbar c}{8\pi G}}}\approx 4{,}340\,\mu \mathrm {g} }.

With the definition of a corresponding reduced Planck charge {\displaystyle {\overline {q_{\mathrm {P} }}}={\sqrt {\frac {\hbar \,c\,\varepsilon _{0}}{2}}}}then the above equality of forces is preserved.

Derived quantities

In addition to these five basic quantities, the following derived quantities are also used:

Name

Size

Dimension

Term

Value in SI units

Planck area

Area

L2

l_\mathrm{P}^2 = \frac{\hbar G}{c^3}

2,612 · 10−70 m2

Planck volume

Volume

L3

l_\mathrm{P}^3 = \sqrt{\frac{\hbar G}{c^3}}^{\,3}

4,222 · 10−105 m3

Planck Energy

Energy

ML2T-2

E_\mathrm{P} = m_\mathrm{P} c^2 = \frac{\hbar}{t_\mathrm{P}} = \sqrt{\frac{\hbar c^5}{G}}

1.956 - 109 J=
1.2209 - 1028 eV=
543.4 kWh

Planck pulse

Impulse

MLT-1

m_\mathrm{P} c = \frac{\hbar}{l_\mathrm{P}} = \sqrt{\frac{\hbar c^3}{G}}

6.525 kg m-s-1

Planck force

Force

MLT-2

F_\mathrm{P} = \frac{E_\mathrm{P}}{l_\mathrm{P}} = \frac{\hbar}{l_\mathrm{P} t_\mathrm{P}} = \frac{c^4}{G}

1,210 · 1044 N

Planck power

Power

ML2T-3

P_\mathrm{P} = \frac{E_\mathrm{P}}{t_\mathrm{P}} = \frac{\hbar}{t_\mathrm{P}^2} = \frac{c^5}{G}

3,628 · 1052 W

Planck density

Density

ML-3

\rho_\mathrm{P} = \frac{m_\mathrm{P}}{l_\mathrm{P}^3} = \frac{\hbar t_\mathrm{P}}{l_\mathrm{P}^5} = \frac{c^5}{\hbar G^2}

5.155 - 1096 kg-m-3

Planck angular frequency

Circular frequency

T−1

\omega_\mathrm{P} = \frac{1}{t_\mathrm{P}} = \sqrt{\frac{c^5}{\hbar G}}

1,855 · 1043 s−1

Planck pressure

Print

ML-1T-2

p_\mathrm{P} = \frac{F_\mathrm{P}}{l_\mathrm{P}^2} = \frac{\hbar}{l_\mathrm{P}^3 t_\mathrm{P}} =\frac{c^7}{\hbar G^2}

4.633 - 10113 Pa

Planck current

Electric current

QT-1

I_\mathrm{P} = \frac{q_\mathrm{P}}{t_\mathrm{P}} = \sqrt{\frac{c^6 4 \pi \varepsilon_0}{G}}

3,479 · 1025 A

Planck voltage

Electrical voltage

ML2T-2Q-1

U_\mathrm{P} = \frac{E_\mathrm{P}}{q_\mathrm{P}} = \frac{\hbar}{t_\mathrm{P} q_\mathrm{P}} = \sqrt{\frac{c^4}{G 4 \pi \varepsilon_0} }

1,043 · 1027 V

Planck impedance

Resistance

ML2T-1Q-2

{\displaystyle Z_{\mathrm {P} }={\frac {U_{\mathrm {P} }}{I_{\mathrm {P} }}}={\frac {\hbar }{q_{\mathrm {P} }^{2}}}={\frac {1}{4\pi \varepsilon _{0}c}}={\frac {Z_{0}}{4\pi }}}

29,98 Ω

Planck acceleration

Acceleration

LT-2

g_\mathrm{P} = \frac{F_\mathrm{P}}{m_\mathrm{P}} = \sqrt{\frac{c^7}{\hbar G}}

5.56 - 1051 m-s-2

Planck magnetic field

Magnetic flux density

MQ-1T-1

{\displaystyle B_{\mathrm {P} }={\sqrt {{\frac {\mu _{0}}{4\pi }}p_{\mathrm {P} }}}={\sqrt {\frac {c^{5}}{\hbar G^{2}4\pi \varepsilon _{0}}}}}

2,1526 · 1053 T

Planck magnetic flux

Magnetic flux

ML2T-1Q-1

{\displaystyle \phi _{\mathrm {P} }={\frac {E_{\mathrm {P} }}{I_{\mathrm {P} }}}={\sqrt {\frac {\hbar }{4\pi \varepsilon _{0}c}}}}

5.6227 - 10-17 Wb

The Planck unit for the angular momentum results from the product of Planck length and Planck momentum to the value \hbar . This is just the unit of angular momentum quantization known from quantum mechanics.

The Planck area l_\mathrm{P}^2plays an important role, in particular, in string theories and in black hole entropy considerations related to the holographic principle.

History

At the end of the 19th century, during his investigations on the theory of radiation of black bodies, for which he received the Nobel Prize in Physics two decades later, Planck discovered the last natural constant necessary for the definition of Planck units, the quantum of action later named after him. He recognized the possibility of using it to define a universally valid system of units and mentioned it in a lecture "On Irreversible Radiation Processes." The following quotation gives an impression of the importance Planck attached to these units

"... the possibility is given to establish units [...] which, independent of special bodies or substances, necessarily retain their meaning for all times and for all, also extraterrestrial and extrahuman cultures, and which therefore can be called 'natural units of measurement'."

- Max Planck

Although Planck dedicated a chapter (§ 159. Natural units of measurement) of his book "Theory of Thermal Radiation" published in 1906 to this system of units and also took up this topic again later, it was not used even within physics. The disadvantages that the value of the gravitational constant was not (and still is) known exactly enough for the use in a system of units, and that practically relevant quantities - expressed in its units - would have absurd numerical values, were not opposed by any advantage, because in no physical theory the quantum of action and the gravitational constant appeared at the same time.

Only after first work on the unification of quantum theory and gravitation in the late 1930s, the later field of application of Planck units emerged. By the time John Archibald Wheeler and Oskar Klein published on the Planck length as the limit of applicability of general relativity in 1955, Planck's proposal had been all but forgotten. After the "rediscovery" of Planck's proposals for such a system of measurements, the name Planck units then became common from 1957.

However, the Planck units in use today differ from Planck's original units because, as quantum mechanics has developed, it has become apparent that \hbar = \frac {h}{2\pi}the more practical natural unit than the hchosen by Planck.

Present meaning

If equations containing the natural constants G, cand \hbar in Planck units, the constants can be omitted. This greatly simplifies the equations in certain disciplines of theoretical physics, such as general relativity, quantum field theories, and the various approaches to quantum gravity.

Planck units also provide an alternative view of the fundamental forces of nature, whose strength is described in the International System of Units(SI) by very different coupling constants. Using the Planck units, the situation is as follows: Between two particles having exactly the Planck mass and the Planck charge, the gravitational force and the electromagnetic force would be exactly equal. The different strength of these forces in our world is the consequence of the fact that a proton and an electron, respectively, have a charge of about 0.085 Planck charges, while their masses are smaller than the Planck mass by 19 and 22 orders of magnitude, respectively. So the question: "Why is gravity so weak?" is equivalent to the question: "Why do the elementary particles have such small masses?

Various physicists and cosmologists deal with the question, whether we could notice, if dimensional physical constants would change slightly, and how the world would look like in case of larger changes. Such speculations have been Gmade, among others, about the speed of light cand the gravitational constant , the latter already since about 1900 in the expansion theory of the earth. The atomic physicist George Gamow means in his popular scientific book Mr. Tompkins in the wonderland that a change of would result in clear changes. c

 

Questions and answers

Q: What are Planck units?

A: Planck units are physical units of measurement first developed by Max Planck, based on four physical constants found in nature. When used to express any of these four physical constants, the value is 1.

Q: What are the four basic Planck units based on?

A: The four basic Planck units are based only on four physical constants found in nature, which include the speed of light in a vacuum (c), the gravitational constant (G), the reduced Planck constant (ħ) and the Boltzmann constant (kB).

Q: Why are they called natural units?

A: They are called natural units because they come only from properties of nature and not from any human construct.

Q: How do natural units help physicists?

A: Natural units help physicists to simplify several recurring algebraic expressions of physical law and reframe questions. They also eliminate human centered arbitrariness from the system of units.

Q: What theories do each of these constants have at least one fundamental physical theory associated with them?

A: c has special relativity associated with it, G has general relativity and Newton's law of universal gravitation associated with it, ħ has quantum mechanics associated with it, ε0 has electrostatics associated with it, and kB has statistical mechanics and thermodynamics associated with it.

Q: Why may Planck Units sometimes be semi-humorously referred to as "God's Units"?

A: They may be referred to as "God's Units" because they eliminate human centered arbitrariness from the system of units and some physicists argue that communication with extraterrestrial intelligence would have to use such a system of units to make common reference to scale.

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